Real-world data distributions often shift continuously across multiple latent factors such as time, geography, and socioeconomic contexts. However, existing domain generalization approaches typically treat domains as discrete or as evolving along a single axis (e.g., time). This oversimplification fails to capture the complex, multidimensional nature of real-world variation. This paper introduces the task of Continuous Domain Generalization (CDG), which aims to generalize predictive models to unseen domains defined by arbitrary combinations of continuous variations. We present a principled framework grounded in geometric and algebraic theories, showing that optimal model parameters across domains lie on a low-dimensional manifold. To model this structure, we propose a Neural Lie Transport Operator (NeuralLio), which enables structure-preserving parameter transitions by enforcing geometric continuity and algebraic consistency. To handle noisy or incomplete domain variation descriptors, we introduce a gating mechanism to suppress irrelevant dimensions and a local chart-based strategy for robust generalization. Extensive experiments on synthetic and real-world datasets, including remote sensing, scientific documents, and traffic forecasting, demonstrate that our method significantly outperforms existing baselines in both generalization accuracy and robustness.

We first introduce some handy concepts and results to make the proof succinct, meanwhile providing more information for understanding our model and theory. We begin with some extended discussions on CSG. Note that a reparameterization unnecessarily has its output dimensions in S, i.e. The condition that p(y|s) = p0(y|ΦS(s,v)) for any v V does not indicate that ΦS(s,v) is constant of v, since p0(y|s0) may ignore the change of s0 = ΦS(s,v) from the change of v. The following lemma shows the meaning of a reparameterization: it allows a CSG to vary while inducing the same distribution on the observed data variables (x,y) (i.e., holding the same effect on describing data). We can now define and verify an equivalent relation on CSGs so that the resulting equivalent class contains CSGs that induce the same (x,y) data distribution and hold the same semantic information in their svariables. We say two CSGs pand p0 are semantic-equivalent, if there exists a homeomorphism11 Φ on S V, such that (i) is semantic-preserving: its output dimensions in S is constant of v, ΦS(s,v) = ΦS(s) for any v V, and (ii) it acts as a reparameterization from p to p0: Φ#[ps,v] = p0s,v, p(x|s,v) = p0(x|Φ(s,v)) and p(y|s) = p0(y|ΦS(s)). A.1 below shows that the defined binary relation is indeed an equivalence relation in common cases. As a reparameterization, Φ allows the two models to have different latent-variable parameterizations while inducing the same distribution on the observed data variables (x,y) (Lemma 9). This definition of semantic-equivalence can be rephrased as the existence of a semantic-preserving reparameterization. With proper model assumptions, we can show that any reparameterization between two CSGs is semantic-preserving, so that semantic-preserving CSGs cannot be converted to each other by a reparameterization that mixes swith v. Lemma 11. For two CSGs pand p0, if p0(y|s) has a statistics M0(s) that is an injective function of s, then any reparameterization Φ from pto p0, if exists, has its ΦS constant of v. Proof. Then the condition that p(y|s) = p0(y|ΦS(s,v)) for any v V indicates that M(s) = M0(ΦS(s,v)). If there exist s S and v(1) 6= v(2) V such that ΦS(s,v(1)) 6= ΦS(s,v(2)), then M0(ΦS(s,v(1))) 6= M0(ΦS(s,v(2))) 11A transformation is a homeomorphism if it is a continuous bijection with continuous inverse. This violates M(s) = M0(ΦS(s,v)) which requires both M0(ΦS(s,v(1))) and M0(ΦS(s,v(2))) to be equal to M(s). We then introduce two mathematical facts. Let z be a random variable on a Euclidean space RdZ with density function pz(z), and let Φ be a homeomorphism on RdZ whose inverse Φ 1 is differentiable.

Conventional supervised learning methods, especially deep ones, are found to be sensitive to out-of-distribution (OOD) examples, largely because the learned representation mixes the semantic factor with the variation factor due to their domain-specific correlation, while only the semantic factor causes the output. To address the problem, we propose a Causal Semantic Generative model (CSG) based on a causal reasoning so that the two factors are modeled separately, and develop methods for OOD prediction from a single training domain, which is common and challenging. The methods are based on the causal invariance principle, with a novel design in variational Bayes for both efficient learning and easy prediction. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor by fitting training data, and this semantic-identification guarantees the boundedness of OOD generalization error and the success of adaptation. Empirical study shows improved OOD performance over prevailing baselines.

Theoretically, we prove that SFB can learn an asymptotically-optimal predictor without test-domain labels. Empirically, we demonstrate the effectiveness of SFB on real and synthetic data.

This view of the data generating process is also adopted and promoted by popular existing works.

Popular models for predicting the output (or label, response, outcome)yfrom theinput (orcovariate)xhavebeenfound erroneous when confronted with a distribution change, even from an essentially irrelevant perturbation like a position shift or background change forimages [91,6,102,41,2,27].